Question

Two distinct numbers $a$ and $b$ are chosen randomly from the set $\{2,2^2,2^3,2^4,.....,2^{25}\}$. Find the probability that ${\log }_{a}b$ is an integer

• A. $\frac{28}{300}$
• B. $\frac{27}{300}$
• C. $\frac{31}{300}$
• D. $\frac{29}{300}$

Answer 1

Answer: (C)

$\frac{31}{300}$

Let $a=2^x$ & $b=2^y\Rightarrow lo{g}_{a}b=\frac{y}{x}$ so

$\frac{y}{x}$ must be integer. We know $x,y\in \{1,2,.....,25\}$

as $y=nx$ so $y$ is multiple of $x$

If $x$

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 to 12, 13 to 25

No. of possible values of y

24, 11, 7, 5, 4, 3, 2, 2, 1, 1

each no number

$\therefore$ required probability $=\frac{62}{25.24}=\frac{31}{300}$